Fractional Fourier Transform

    The Quadratic Function in the Fractional Fourier Transform Kernel

    Demonstration 4: Combined View with 3D-Surface Plot

    The Fractional Fourier Transform

     

    kernel comprises an exponential function with argument that is hyperbolic with respect to x and y:

    This can be shown to be a hyperbola whose asymptotes counter-rotate with the power α as it increases through the internal [0, 4) and repeating periodically.

    shows an animation of the counter-rotating asymptotes superimposed upon the contour plot of hyperbolic quadratic form as the power α cycles though its periodic range.

    This demonstration shows a three-dimensional surface plot of kernel's hyperbolic quadratic form as the power α cycles though its periodic range. Here the colors depict synthetic "optical reflections" rendered on the surface by Mathematica's default spatially-distributed illumination sources of red, green and blue to enhance visual queues in the 3D effect.

    Note the alternating antipodal "flapping" of the surface as the kernel approaches and becomes a delta function (as the hyperbolic quadratic form degenerates into a line) at even-integer values of α , as described on the previous three web demonstration pages in this series. Note the two renderings of scale, differing by a factor of 100.

    For even-integer values of , the hyperbola degenerates into the line:

    x-y = 0 for ,

    (that is, where even-integer multiples of 2) or

    x+y = 0 for

    (that is, where odd-integer multiples of 2).

    At either of these even-integer values of , the fractional Fourier transform kernel becomes one of two types of delta function (in the sense of distributions within Schwartz Space extension of L2 )

     
    (Identity operator)
    (Reflection operator)

      REFERENCES

      [1] ... roots and zeros of hyperbolic behavior